L1/LDG Method for Caputo-Hadamard Time Fractional Diffusion Equation

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Partial Differential Equations

Background:

• Fractional diffusion equations model complex physical phenomena.
• Existing numerical methods may face challenges with robustness, especially for small time steps.
• The Caputo-Hadamard fractional derivative is a key component in certain diffusion models.

Purpose of the Study:

• To propose a new class of discrete Gronwall inequalities.
• To apply these inequalities to analyze L1/local discontinuous Galerkin (LDG) finite element methods.
• To demonstrate the $$-robustness of the numerical methods for the Caputo-Hadamard time fractional diffusion equation.

Main Methods:

• Development of discrete Gronwall inequalities.
• Analysis of L1/local discontinuous Galerkin (LDG) finite element methods.
• Theoretical analysis of numerical scheme stability and convergence.

Main Results:

• A novel class of discrete Gronwall inequalities was established.
• The L1/LDG finite element methods were analyzed using the new inequalities.
• The numerical methods were proven to be $$-robust, maintaining validity for small $$ values.

Conclusions:

• The proposed discrete Gronwall inequalities are effective tools for analyzing numerical methods for fractional diffusion equations.
• The L1/LDG methods are robust and suitable for solving the Caputo-Hadamard time fractional diffusion equation.
• Theoretical findings are supported by numerical experiments.